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Regression Approach To ANOVA

- Dummy (Indicator) Variables: Variables that take on the value 1 if observation comes from a particular group, 0 if not.
- If there are I groups, we create I-1 dummy variables.
- Individuals in the “baseline” group receive 0 for all dummy variables.
- Statistical software packages typically assign the “last” (Ith) category as the baseline group
- Statistical Model: E(Y) = b0 + b1Z1+ ... + bI-1ZI-1
- Zi =1 if observation is from group i, 0 otherwise
- Mean for group i (i=1,...,I-1): mi = b0 + bi
- Mean for group I: mI = b0

2-Way ANOVA

- 2 nominal or ordinal factors are believed to be related to a quantitative response
- Additive Effects: The effects of the levels of each factor do not depend on the levels of the other factor.
- Interaction: The effects of levels of each factor depend on the levels of the other factor
- Notation: mij is the mean response when factor A is at level i and Factor B at j

Example - Thalidomide for AIDS

- Response: 28-day weight gain in AIDS patients
- Factor A: Drug: Thalidomide/Placebo
- Factor B: TB Status of Patient: TB+/TB-
- Subjects: 32 patients (16 TB+ and 16 TB-). Random assignment of 8 from each group to each drug). Data:
- Thalidomide/TB+: 9,6,4.5,2,2.5,3,1,1.5
- Thalidomide/TB-: 2.5,3.5,4,1,0.5,4,1.5,2
- Placebo/TB+: 0,1,-1,-2,-3,-3,0.5,-2.5
- Placebo/TB-: -0.5,0,2.5,0.5,-1.5,0,1,3.5

ANOVA Approach

- Total Variation (SST) is partitioned into 4 components:
- Factor A: Variation in means among levels of A
- Factor B: Variation in means among levels of B
- Interaction: Variation in means among combinations of levels of A and B that are not due to A or B alone
- Error: Variation among subjects within the same combinations of levels of A and B (Within SS)

ANOVA Calculations

- Balanced Data (n observations per treatment)

ANOVA Approach

General Notation: Factor A has I levels, B has J levels

- Procedure:
- Test H0: No interaction based on the FAB statistic
- If the interaction test is not significant, test for Factor A and B effects based on the FA and FB statistics

Example - Thalidomide for AIDS

- There is a significant Drug*TB interaction (FDT=5.897, P=.022)
- The Drug effect depends on TB status (and vice versa)

Regression Approach

- General Procedure:
- Generate I-1 dummy variables for factor A (A1,...,AI-1)
- Generate J-1 dummy variables for factor B (B1,...,BJ-1)
- Additive (No interaction) model:

Tests based on fitting full and reduced models.

Example - Thalidomide for AIDS

- Factor A: Drug with I=2 levels:
- D=1 if Thalidomide, 0 if Placebo
- Factor B: TB with J=2 levels:
- T=1 if Positive, 0 if Negative
- Additive Model:
- Population Means:
- Thalidomide/TB+: b0+b1+b2
- Thalidomide/TB-: b0+b1
- Placebo/TB+: b0+b2
- Placebo/TB-: b0
- Thalidomide (vs Placebo Effect) Among TB+/TB- Patients:
- TB+: (b0+b1+b2)-(b0+b2) = b1 TB-: (b0+b1)- b0 = b1

Example - Thalidomide for AIDS

- Testing for a Thalidomide effect on weight gain:
- H0: b1 = 0 vs HA: b1 0 (t-test, since I-1=1)
- Testing for a TB+ effect on weight gain:
- H0: b2 = 0 vs HA: b2 0 (t-test, since J-1=1)
- SPSS Output: (Thalidomide has positive effect, TB None)

Regression with Interaction

- Model with interaction (A has I levels, B has J):
- Includes I-1 dummy variables for factor A main effects
- Includes J-1 dummy variables for factor B main effects
- Includes (I-1)(J-1) cross-products of factor A and B dummy variables
- Model:

As with the ANOVA approach, we can partition the variation to that attributable to Factor A, Factor B, and their interaction

Example - Thalidomide for AIDS

- Model with interaction:E(Y)=b0+b1D+b2T+b3(DT)
- Means by Group:
- Thalidomide/TB+: b0+b1+b2+b3
- Thalidomide/TB-: b0+b1
- Placebo/TB+: b0+b2
- Placebo/TB-: b0
- Thalidomide (vs Placebo Effect) Among TB+ Patients:
- (b0+b1+b2+b3)-(b0+b2) = b1+b3
- Thalidomide (vs Placebo Effect) Among TB- Patients:
- (b0+b1)-b0= b1
- Thalidomide effect is same in both TB groups if b3=0

Example - Thalidomide for AIDS

- SPSS Output from Multiple Regression:

We conclude there is a Drug*TB interaction (t=2.428, p=.022). Compare this with the results from the two factor ANOVA table

Repeated Measures ANOVA

- Goal: compare g treatments over t time periods
- Randomly assign subjects to treatments (Between Subjects factor)
- Observe each subject at each time period (Within Subjects factor)
- Observe whether treatment effects differ over time (interaction, Within Subjects)

Repeated Measures ANOVA

- Suppose there are N subjects, with ni in the ith treatment group.
- Sources of variation:
- Treatments (g-1 df)
- Subjects within treatments aka Error1 (N-g df)
- Time Periods (t-1 df)
- Time x Trt Interaction ((g-1)(t-1) df)
- Error2 ((N-g)(t-1) df)

Repeated Measures ANOVA

To Compare pairs of treatment means (assuming no time by treatment interaction, otherwise they must be done within time periods and replace tn with just n):

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